PLANE CURVES AND FREE HAND SKETCHING

CONSTRUCTION OF PARALLEL & PERPENDICULAR LINES

  1. 1.Given are the steps to draw a perpendicular
    to a line at a point within the line when the
    point is near the Centre of a line.

    Arrange the steps. Let AB be the line and P
    be the point in it
    i. P as Centre, take convenient radius R1 and
    draw arcs on the two sides of P on the line at
    C, D.
    ii. Join E and P
    iii. The line EP is perpendicular to AB
    iv. Then from C, D as Centre, take R2 radius
    (greater than R1), draw arcs which cut at E.
    a) i, iv, ii, iii
    b) iii, ii, iv, i
    c) iv, iii, i, ii
    d) ii, i, iv, iii
    Answer: a

Explanation: Here uses the concept of a locus. Every 2 points have a particular line
that is every point on line is equidistant from
both the points. The above procedure shows
how the line is build up using arcs of the
similar radius.

  1. 2.Given are the steps to draw a perpendicular
    to a line at a point within the line when the
    point is near an end of the line.
    Arrange the steps. Let AB be the line and P
    be the point in it.

    i. Join the D and P.
    ii. With any point O draw an arc (more than a
    semicircle) with a radius of OP, cuts AB at C.
    iii. Join the C and O and extend till it cuts the
    large arc at D.
    iv. DP gives the perpendicular to AB.
    a) i, iv, ii, iii
    b) iii, ii, iv, i
    c) iv, iii, i, ii
    d) ii, iii, i, iv
    Answer: d
    Explanation: There exists a common
    procedure for obtaining perpendiculars for
    lines. But changes are due changes in
    conditions whether the point lies on the line,
    off the line, near the centre or near the ends
    etc.
  2. 3.Given are the steps to draw a perpendicular
    to a line at a point within the line when the
    point is near the centre of line.
    Arrange the steps. Let AB be the line and P
    be the point in it

    i. Join F and P which is perpendicular to AB.
    ii. Now C as centre take the same radius and
    cut the arc at D and again D as centre with
    same radius cut the arc further at E.
    iii. With centre as P take any radius and draw
    an arc (more than a semicircle) cuts AB at C.
    iv. Now D, E as centre take radius (more than
    half of DE) draw arcs which cut at F.
    a) i, iv, ii, iii
    b) iii, ii, iv, i
    c) iv, iii, i, ii
    d) ii, i, iv, iii

Answer: b
Explanation: Generally in drawing
perpendiculars to lines involves in drawing a
line which gives equidistance from either side
of the line to the base, which is called the
locus of points. But here since the point P is
nearer to end, there exists some peculiar steps
in drawing arcs.

  1. 4.Given are the steps to draw a perpendicular
    to a line from a point outside the line, when
    the point is near the centre of line.
    Arrange the steps. Let AB be the line and P
    be the point outside the line

    i. The line EP is perpendicular to AB
    ii. From P take convenient radius and draw
    arcs which cut AB at two places, say C, D.
    iii. Join E and P.
    iv. Now from centers C, D draw arc with
    radius (more than half of CD), which cut each
    other at E.
    a) i, iv, ii, iii
    b) iii, ii, iv, i
    c) iv, iii, i, ii
    d) ii, iv, iii, i
    Answer: d
    Explanation: At first two points are taken
    from the line to which perpendicular is to
    draw with respect to P. Then from two points
    equidistant arcs are drawn to meet at some
    point which is always on the perpendicular.
    So by joining that point and P gives
    perpendicular.
  1. 5.Given are the steps to draw a perpendicular
    to a line from a point outside the line, when
    the point is near an end of the line.

    Arrange the steps. Let AB be the line and P
    be the point outside the line

    i. The line ED is perpendicular to AB
    ii. Now take C as centre and CP as radius cut
    the previous arc at two points say D, E.
    iii. Join E and D.
    iv. Take A as center and radius AP draw an
    arc (semicircle), which cuts AB or extended
    AB at C.
    a) i, iv, ii, iii
    b) iii, ii, iv, i
    c) iv, ii, i, iii
    d) ii, iv, iii, i
    Answer: c
    Explanation: The steps here show how to
    draw a perpendicular to a line from a point
    when the point is nearer to end of line. Easily
    by drawing arcs which are equidistance from
    either sides of line and coinciding with point
    P perpendicular has drawn.
  2. 6.Given are the steps to draw a perpendicular
    to a line from a point outside the line, when
    the point is nearer the centre of line.
    Arrange the steps. Let AB be the line and P
    be the point outside the line

    i. Take P as centre and take some convenient
    radius draw arcs which cut AB at C, D.
    ii. Join E, F and extend it, which is
    perpendicular to AB.
    iii. From C, D with radius R1 (more than half
    of CD), draw arcs which cut each other at E.
    iv. Again from C, D with radius R2 (more
    than R1), draw arcs which cut each other at F.
    a) i, iii, iv, ii
    b) iii, ii, iv, i
    c) iv, iii, i, ii
    d) ii, iv, iii, i
    Answer: a
    Explanation: For every two points there
    exists a line which has points from which
    both the points are equidistant otherwise
    called perpendicular to line joining the two
    points. Here at 1st step, we created two on the
    line we needed perpendicular, then with equal
    arcs from either sides we created the
    perpendicular.
  1. 7.Given are the steps to draw a parallel line
    to given line AB at given point P.
    Arrange the steps.

    i. Take P as centre draw a semicircle which
    cuts AB at C with convenient radius.
    ii. From C with radius of PD draw an arc with
    cuts the semicircle at E.
    iii. Join E and P which gives parallel line to AB. iv. From C with same radius cut the AB at D.

a) i, iv, ii, iii
b) iii, ii, iv, i
c) iv, iii, i, ii
d) ii, iv, iii, i
Answer: a
Explanation: There exists some typical steps
in obtaining parallel lines for required lines at
given points which involves drawing of arcs,
necessarily, here to form a parallelogram
since the opposite sides in parallelogram are
parallel.

  1. 8.Given are the steps to draw a parallel line
    to given line AB at a distance R.
    Arrange the steps.

    i. EF is the required parallel line.
    ii. From C, D with radius R, draw arcs on the
    same side of AB.
    iii. Take two points say C, D on AB as far as
    possible.
    iv. Draw a line EF which touches both the arc
    (tangents) at E, F.
    a) i, iv, ii, iii
    b) iii, ii, iv, i
    c) iv, iii, i, ii
    d) ii, iv, iii, i
    Answer: b
    Explanation: Since there is no reference
    point P to draw parallel line, but given the
    distance, we can just take arcs with distance
    given from the base line and draw tangent
    which touches both arcs.
  2. 9.Perpendiculars can’t be drawn using
    a) T- Square
    b) Set-squares
    c) Pro- circle
    d) Protractor
    Answer: c
    Explanation: T-square is meant for drawing a straight line and also perpendiculars. And also using set-squares we can draw perpendiculars. Protractor is used to measure angles and also we can use to draw perpendiculars. But pro-circle consists of circles of different diameters.
  1. 10.The length through perpendicular gives
    the shortest length from a point to the line.

    a) True
    b) False
    Answer: a
    Explanation: The statement given here is
    right. If we need the shortest distance from a
    point to the line, then drawing perpendicular
    along the point to a line is the best method.
    Since the perpendicular is the line which has
    points equidistant from points either side of
    given line.

DRAWING REGULAR POLYGONS & SIMPLE CURVES

  1. 1.A Ogee curve is a
    a) semi ellipse
    b) continuous double curve with convex and
    concave
    c) freehand curve which connects two parallel
    lines
    d) semi hyperbola
    Answer: b
    Explanation: An ogee curve or a reverse
    curve is a combination of two same curves in
    which the second curve has a reverse shape to
    that of the first curve. Any curve or line or
    mould consists of a continuous double curve
    with the upper part convex and lower part
    concave, like ‘’S’’.
  1. 2.Given are the steps to construct an
    equilateral triangle, when the length of side is
    given. Using, T-square, set-squares only.
    Arrange the steps.

    i. The both 2 lines meet at C. ABC is required
    triangle
    ii. With a T-square, draw a line AB with
    given length

iii. With 30o-60o set-squares, draw a line
making 60o with AB at A
iv. With 30o-60o set-squares, draw a line
making 60o with AB at B
a) i, iv, ii, iii
b) iii, ii, iv, i
c) iv, iii, i, ii
d) ii, iii, iv, i
Answer: d
Explanation: Here gives the simple
procedure since T-square and 30o
-60o setsquares. And also required triangle is
equilateral triangle. The interior angles are
60°, 60°, 60° (180° /3 = 60°). Set- squares are
used for purpose of 60°.

  1. 3.Given are the steps to construct an
    equilateral triangle, with help of a compass,
    when the length of a side is given. Arrange
    the steps.

    i. Draw a line AB with given length
    ii. Draw lines joining C with A and B
    iii. ABC is required equilateral triangle
    iv. With centers A and B and radius equal to
    AB, draw arcs cutting each other at C
    a) i, iv, ii, iii
    b) iii, ii, iv, i
    c) iv, iii, i, ii
    d) ii, iii, iv, i
    Answer: a
    Explanation: Here gives the simple
    procedure to construct an equilateral triangle.
    Since we used compass we can construct any
    type of triangle but with set-squares it is not
    possible to construct any type of triangles
    such as isosceles, scalene etc.
  2. 4.Given are the steps to construct an
    equilateral triangle when the altitude of a
    triangle is given. Using, T-square, set-squares
    only. Arrange the steps.

    i. Join R, Q; T, Q. Q, R, T is the required
    triangle
    ii. With a T-square, draw a line AB of any
    length
    iii. From a point P on AB draw a
    perpendicular PQ of given altitude length
    iv. With 30o-60o set-squares, draw a line
    making 30o with PQ at Q on both sides
    cutting at R, T
    a) i, iv, ii, iii
    b) iii, ii, iv, i
    c) iv, iii, i, ii
    d) ii, iii, iv, i
    Answer: d
    Explanation: Here gives the simple
    procedure since T-square and 30°-60°setsquares. The interior angles are 60°, 60°, 60°
    (180° /3 = 60°). Altitude divides the sides of equilateral triangle equally. Set- squares are
    used for purpose of 30°.
  3. 5.Given are the steps to construct an
    equilateral triangle, with help of a compass,
    when the length of altitude is given. Arrange
    the steps.

    i. Draw a line AB of any length. At any point
    P on AB, draw a perpendicular PQ equal to
    altitude length given
    ii. Draw bisectors of CE and CF to intersect
    AB at R and T respectively.QRT is required
    triangle
    iii. With center Q and any radius, draw an arc
    intersecting PQ at C
    iv. With center C and the same radius, draw
    arcs cutting the 1st arc at E and F
    a) i, iii, iv, ii
    b) iii, ii, iv, i
    c) iv, iii, i, ii
    d) ii, iii, iv, i
    Answer: a
    Explanation: This is the particular procedure
    used for only constructing an equilateral
    triangle using arcs when altitude is given
    since we used similar radius arcs to get 30o
    on both sides of a line. Here we also bisected
    arc using the same procedure from bisecting
    lines.
  1. 6.How many pairs of parallel lines are there
    in regular Hexagon?

    a) 2
    b) 3
    c) 6
    d) 1
    Answer: b
    Explanation: Hexagon is a closed figure
    which has six sides, six corners. Given is
    regular hexagon which means it has equal
    interior angles and equal side lengths. So,
    there will be 3 pair of parallel lines in a
    regular hexagon.
  2. 7.Given are the steps to construct a square when the length of a side is given. Using, T square, set-squares only. Arrange the steps.
    i. Repeat the previous step and join A, B, C
    and D to form a square
    ii. With a T-square, draw a line AB with
    given length.
    iii. At A and B, draw verticals AE and BF
    iv. With 45o set-squares, draw a line making
    45o with AB at A cuts BF at C
    a) i, iv, ii, iii
    b) iii, ii, iv, i
    c) iv, iii, i, ii
    d) ii, iii, iv, i
    Answer: d
    Explanation: Square is closed figure with
    equal sides and equal interior angles which is
    90°. In the above steps, it is given the
    procedure to draw a square using set-squares.
    45° set-square is used since 90/2 = 45.
  3. 8.How many pairs of parallel lines are there
    in a regular pentagon?

    a) 0
    b) 1
    c) 2
    d) 5
    Answer: a
    Explanation: Pentagon is a closed figure
    which has five sides, five corners. Given is
    regular pentagon which means it has equal
    interior angles and equal side lengths. Since
    five is odd number so, there exists angles 36°,
    72°, 108°, 144°, 180° with sides to
    horizontal.
  4. 9.Given are the steps to construct a square
    using a compass when the length of the side
    is given. Arrange the steps.

    i. Join A, B, C and D to form a square
    ii. At A with radius AB draw an arc, cut the
    AE at D
    iii. Draw a line AB with given length. At A
    draw a perpendicular AE to AB using arcs
    iv. With centers B and D and the same radius,
    draw arcs intersecting at C
    a) i, iv, ii, iii
    b) iii, ii, iv, i
    c) iv, iii, i, ii
    d) ii, iii, iv, i
    Answer: b
    Explanation: Here we just used simple
    techniques like drawing perpendiculars using
    arcs and then used the compass to locate the
    fourth point. Using the compass it is easier to
    draw different types of closed figures than
    using set-squares.
  1. 10.Given are the steps to construct regular
    polygon of any number of sides. Arrange the
    steps.

    i. Draw the perpendicular bisector of AB to
    cut the line AP in 4 and the arc AP in 6
    ii. The midpoint of 4 and 6 gives 5 and
    extension of that line along the equidistant
    points 7, 8, etc gives the centers for different
    polygons with that number of sides and the
    radius is AN (N is from 4, 5, 6, 7, so on to N)
    iii. Join A and P. With center B and radius
    AB, draw the quadrant AP
    iv. Draw a line AB of given length. At B,
    draw a line BP perpendicular and equal to AB
    a) i, iv, ii, iii
    b) iii, ii, iv, i

c) iv, iii, i, ii
d) ii, iii, iv, i
Answer: c
Explanation: Given here is the method for
drawing regular polygons of a different
number of sides of any length. This includes
finding a line where all the centers for regular
polygons lies and then with radius taking any
end of 1st drawn line to center and then
completing circle at last, cutting the circle
with the same length of initial line. Thus we
acquire polygons.

DRAWING TANGENTS AND NORMALS FOR DIFFERENT CONDITIONS OF CIRCLE

  1. 1.Given are the steps to draw a tangent to
    any given circle at any point P on it. Arrange
    the steps.

    i. Draw the given circle with center O and
    mark the point P anywhere on the circle.
    ii. With centers O and Q draw arcs with equal
    radius to cut each other at R.
    iii. Join R and P which is the required
    tangent.
    iv. Draw a line joining O and P. Extend the
    line to Q such that OP = PQ.
    a) i, iv, ii, iii
    b) iv, i, iii, ii
    c) iii, i, iv, ii
    d) ii, iv, i, iii
    Answer: a
    Explanation:
    Tangent is a line which touches
    a curve at only one point. Every tangent is
    perpendicular to its normal. Here we first
    found the normal which passes through center
    and point. Then drawing a perpendicular to it
    gives the tangent.
  2. 2.Given are the steps to draw a tangent to
    given circle from any point outside the circle.
    Arrange the steps.

    i. With OP as diameter, draw arcs on circle at
    R and R1.
    ii. Draw the given circle with center O.
    iii. Join P and R which is one tangent and
    PR1 is another tangent.
    iv. Mark the point P outside the circle.
    a) ii, iv, iii, i
    b) iv, i, iii, ii
    c) iii, i, iv, ii
    d) ii, iv, i, iii
    Answer: d
    Explanation:
    Usually when a point is outside
    the circle there exists two tangents. For which
    we first join the center with point P and then
    taking distance from center to P as diameter
    circle is drawn from the midpoint of center
    and P to cut circle at two points where
    tangents touch the circle.
  3. 3.Given are the steps to draw a tangent to
    given arc even if center is unknown and the
    point P lies on it. Arrange the steps. Let AB
    be the arc.

    i. Draw EF, the bisector of the arc CD. It will
    pass through P.
    ii. RS is the required tangent.
    iii. With P as center and any radius draw arcs
    cutting arc AB at C and D.
    iv. Draw a perpendicular RS to EF through P.
    a) ii, iv, iii, i
    b) iv, i, iii, ii
    c) iii, i, iv, ii
    d) ii, iv, i, iii
    Answer: c
    Explanation:
    Even if the center of the arc is
    unknown, just by taking any some part of arc
    and bisecting that with a line at required point
    p gives us normal to tangent at P. So then
    from normal drawing perpendicular gives our
    required tangent.
  4. 4.Given are the steps to draw a tangent to
    given circle and parallel to given line.
    Arrange the steps.

    i. Draw a perpendicular to given line and
    extend to cut the circle at two points P and Q

ii. At P or Q draw perpendicular to normal
then we get the tangents.
iii. PQ is the normal for required tangent.
iv. Draw a circle with center O and line AB as
required.
a) ii, iv, iii, i
b) iv, i, iii, ii
c) iii, i, iv, ii
d) ii, iv, i, iii
Answer: b
Explanation:
Normal of curve will be
perpendicular to every parallel tangent at that
point. We just drawn the longest chord
(diameter) and then perpendicular it gives the
required tangents. Since circle is closed figure
there exist two tangents parallel to each other.

  1. 5.How many external tangents are there for
    two circles?

    a) 1
    b) 2
    c) 3
    d) 4
    Answer: b
    Explanation:
    External tangents are those
    which touch both the circles but they will not
    intersect in between the circles. The tangents
    touch at outmost points of circles that are
    ends of diameter if the circles have the same
    diameter.
  2. 6.How many internal tangents are there for
    two circles?

    a) 4
    b) 3
    c) 2
    d) 1
    Answer: c
    Explanation:
    Internal tangents are those
    which touch both the circle and also intersect
    each other on the line joining the centers of
    circles. And the internal tangents intersect
    each other at midpoint of line joining the
    center of circles only if circles have the same
    diameter.
  1. 7.For any point on any curve there exist two
    normals.

    a) True
    b) False
    Answer: b
    Explanation:
    Here we take point on the
    curve. There exist multiple tangents for some
    curve which are continuous, trigonometric
    curves, hyperbola etc. But for curves like
    circles, parabola, ellipse, cycloid etc. have
    only one tangent and normal.
  2. 8.Arrange the steps. These give procedure to
    draw internal tangent to two given circles of
    equal radii.

    i. Draw a line AB which is the required
    tangent.
    ii. Draw the given circles with centers O and
    P.
    iii. With center R and radius RA, draw an arc
    to intersect the other circle on the other circle
    on the other side of OP at B.
    iv. Bisect OP in R. Draw a semi circle with
    OR as diameter to cut the circle at A.
    a) ii, iv, iii, i
    b) iv, i, iii, ii
    c) iii, i, iv, ii
    d) ii, iv, i, iii
    Answer: a
    Explanation:
    Since the circles have same
    radius. The only two internal tangents will
    intersect at midpoint of line joining the
    centers. So we first found the center and then
    point of intersection of tangent and circle then
    from that point to next point it is drawn a arc
    midpoint as center and join the points gave us
    tangent.
  1. 9.There are 2 circles say A, B. A has 20 units
    radius and B has 10 units radius and distance
    from centers of A and B is 40 units. Where
    will be the intersection point of external
    tangents?

    a) to the left of two circles
    b) to the right of the two circles
    c) middle of the two circles
    d) they intersect at midpoint of line joining
    the centers
    Answer: b
    Explanation:
    A has 20 units radius and B has
    10 units radius. So, the tangents go along the
    circles and meet at after the second circle that
    is B that is the right side of both circles. And
    we asked for external tangents so they meet
    away from the circles but not in between
    them.

10.10.There are 2 circles say A, B. A is smaller
than B and they are not intersecting at any
point. Where will be the intersection point of
internal tangents for these circles?

a) to the left of two circles
b) to the right of the two circles
c) middle of the two circles
d) they intersect at midpoint of line joining
the centers
Answer: b
Explanation:
A is smaller than B so the
intersection point of internal tangents will not
be on the midpoint of the line joining the
centers. And we asked for internal tangents so
they will not meet away from the circles.

They meet in between them

CONSTRUCTION OF ELLIPSE – 1

  1. 1.Which of the following is incorrect about
    Ellipse?

    a) Eccentricity is less than 1
    b) Mathematical equation is X2 /a2 + Y2/b2 =1
    c) If a plane is parallel to axis of cone cuts the
    cone then the section gives ellipse
    d) The sum of the distances from two focuses
    and any point on the ellipse is constant
    Answer: c
    Explanation: If a plane is parallel to the axis
    of cone cuts the cone then the cross-section
    gives hyperbola. If the plane is parallel to
    base it gives circle. If the plane is inclined
    with an angle more than the external angle of
    cone it gives parabola. If the plane is inclined
    and cut every generators then it forms an
    ellipse.
  2. 2.Which of the following constructions
    doesn’t use elliptical curves?

    a) Cooling towers
    b) Dams
    c) Bridges
    d) Man-holes
    Answer: a
    Explanation:
    Cooling towers, water channels use Hyperbolic curves as their design, Arches, Bridges, sound reflectors, lighter flectors etc use parabolic curves. Arches, bridges, dams, monuments, man-holes, glandsbridges, dams, monuments, man-holes, glands and stuffing boxes etc use elliptical curves.
  1. 3.The line which passes through the focus
    and perpendicular to the major axis is

    a) Minor axis
    b) Latus rectum
    c) Directrix
    d) Tangent
    Answer: b
    Explanation:
    The line bisecting the major
    axis at right angles and terminated by curve is
    called the minor axis. The line which passes
    through the focus and perpendicular to the
    major axis is latus rectum. Tangent is the line
    which touches the curve at only one point.
  2. 4.Which of the following is the eccentricity
    for an ellipse?

    a) 1
    b) 3/2
    c) 2/3
    d) 5/2
    Answer: c
    Explanation:
    The eccentricity for ellipse is
    always less than 1. The eccentricity is always
    1 for any parabola. The eccentricity is always
    0 for a circle. The eccentricity for a hyperbola
    is always greater than 1.
  3. 5.Axes are called conjugate axes when they
    are parallel to the tangents drawn at their
    extremes.

    a) True
    b) False
    Answer: a
    Explanation:
    In ellipse there exist two axes
    (major and minor) which are perpendicular to
    each other, whose extremes have tangents
    parallel them. There exist two conjugate axes
    for ellipse and 1 for parabola and hyperbola.
  4. 6.Steps are given to draw an ellipse by loop
    of the thread method. Arrange the steps.

    i. Check whether the length of the thread is
    enough to touch the end of minor axis.
    ii. Draw two axes AB and CD intersecting at
    O. Locate the foci F1 and F2.
    iii. Move the pencil around the foci,
    maintaining an even tension in the thread
    throughout and obtain the ellipse.
    iv. Insert a pin at each focus-point and tie a
    piece of thread in the form of a loop around
    the pins.
    a) i, ii, iii, iv
    b) ii, iv, i, iii
    c) iii, iv, i, ii
    d) iv, i, ii, iii
    Answer: b
    Explanation:
    This is the easiest method of
    drawing ellipse if we know the distance
    between the foci and minor axis, major axis.
    It is possible since ellipse can be traced by a
    point, moving in the same plane as and in
    such a way that the sum of its distances from
    two foci is always the same.
  5. 7.Steps are given to draw an ellipse by
    trammel method. Arrange the steps.
    i. Place the trammel so that R is on the minor
    axis CD and Q on the major axis AB. Then P
    will be on the ellipse.

    ii. Draw two axes AB and CD intersecting
    each other at O.
    iii. By moving the trammel to new positions,
    always keeping R on CD and Q on AB,
    obtain other points and join those to get an
    ellipse.
    iv. Along the edge of a strip of paper which
    may be used as a trammel, mark PQ equal to
    half the minor axis and PR equal to half of
    major axis.
    a) i, ii, iii, iv
    b) ii, iv, i, iii
    c) iii, iv, i, ii
    d) iv, i, ii, iii
    Answer: b
    Explanation:
    This method uses the trammels PQ and PR which ends Q and R should be placed on major axis and minor axis respectively. It is possible since ellipse can be traced by a point, moving in the same plane as and in such a way that the sum of its distances from two foci is always the same.
  1. 8.Steps are given to draw a normal and a
    tangent to the ellipse at a point Q on it.
    Arrange the steps.

    i. Draw a line ST through Q and
    perpendicular to NM.
    ii. ST is the required tangent.
    iii. Join Q with the foci F1 and F2.
    iv. Draw a line NM bisecting the angle
    between the lines drawn before which is
    normal.
    a) i, ii, iii, iv
    b) ii, iv, i, iii
    c) iii, iv, i, ii
    d) iv, i, ii, iii
    Answer: c
    Explanation:
    Tangents are the lines which
    touch the curves at only one point. Normals
    are perpendiculars of tangents. As in the
    circles first, we found the normal using foci
    (centre in circle) and then perpendicular at
    given point gives tangent.
  2. 9.Which of the following is not belonged to
    ellipse?

    a) Latus rectum
    b) Directrix
    c) Major axis
    d) Asymptotes
    Answer: d
    Explanation:
    Latus rectum is the line joining
    one of the foci and perpendicular to the major
    axis. Asymptotes are the tangents which meet
    the hyperbola at infinite distance. Major axis
    consists of foci and perpendicular to the
    minor axis.

CONSTRUCTION OF ELLIPSE – 2

  1. 1.Mathematically, what is the equation of
    ellipse?

    a) x2/a2 + y2/b2 = -1
    b) x2/a2– y2/b2 = 1
    c) x2/a2 + y2/b2 = 1
    d) x2/a2– y2/b2 = 1
    Answer: c
    Explanation:
    Equation of ellipse is given by;
    x2/a2 + y2/b2 = 1. Here, a and b are half the
    distance of lengths of major and minor axes
    of the ellipse. If the value of a = b then the
    resulting ellipse will be a circle with Centre
    (0,0) and radius equal to a units.
  2. 2.In general method of drawing an ellipse, a
    vertical line called as is drawn first.

    a) Tangent
    b) Normal
    c) Major axis
    d) Directrix
    Answer: d
    Explanation:
    In the general method of
    drawing an ellipse, a vertical line called as
    directrix is drawn first. The focus is drawn at
    a given distance from the directrix drawn.
    The eccentricity of the ellipse is less than one.
  3. 3.If eccentricity of ellipse is 3/7, how many
    divisions will the line joining the directrix
    and the focus have in general method?

    a) 10
    b) 7
    c) 3
    d) 5
    Answer: a
    Explanation
    : In the general method of
    drawing an ellipse, if eccentricity of the
    ellipse is given as 3/7 then the line joining the
    directrix and the focus will have 10 divisions.
    The number is derived by adding the
    numerator and denominator of the
    eccentricity
  1. 4.In the general method of drawing an
    ellipse, after parting the line joining the
    directrix and the focus, a is made.

    a) Tangent
    b) Vertex
    c) Perpendicular bisector
    d) Normal
    Answer: b
    Explanation:
    In the general method of
    drawing after parting the line joining the
    directrix and the focus, a vertex is made. An
    arc with a radius equal to the length between
    the vertex and the focus is drawn with the
    vertex as the centre.
  2. 5.An ellipse is defined as a curve traced by a
    point which has the sum of distances between
    any two fixed points always same in the same
    plane.

    a) True
    b) False
    Answer: a
    Explanation:
    An ellipse can also be defined
    as a curve that can be traced by a point
    moving in the same plane with the sum of the
    distances between any two fixed points
    always same. The two fixed points are called
    as a focus.
  3. 6.An ellipse has foci.
    a) 1
    b) 2
    c) 3
    d) 4
    Answer: b
    Explanation:
    An ellipse has 2 foci. These
    foci are fixed in a plane. The sum of the
    distances of a point with the foci is always
    same. The ellipse can also be defined as the
    curved traced by the points which exhibit this
    property.
  4. 7.If information about the major and minor
    axes of ellipse is given then by how many
    methods can we draw the ellipse?

    a) 2
    b) 3
    c) 4
    d) 5
    Answer: d
    Explanation:
    There are 5 methods by which
    we can draw an ellipse if we know the major
    and minor axes of that ellipse. Those five
    methods are arcs of circles method,
    concentric circles method, loop of the thread
    method, oblong method, trammel method.
  5. 8.In arcs of circles method, the foci are
    constructed by drawing arcs with centre as
    one of the ends of the axis and the
    radius equal to the half of the axis.

    a) Minor, major
    b) Major, major
    c) Minor, minor
    d) Major, minor
    Answer: a
    Explanation:
    In arcs of circles method, the
    foci are constructed by drawing arcs with
    centre as one of the ends of the minor axis
    and the radius equal to the half of the major
    axis. This method is used when we know only
    major and minor axes of the ellipse.
  6. 9.If we know the major and minor axes of
    the ellipse, the first step of drawing the
    ellipse, we draw the axes each other.

    a) Parallel to
    b) Perpendicular bisecting
    c) Just touching
    d) Coinciding
    Answer: b
    Explanation:
    If we know the major and
    minor axes of the ellipse, the first step of the
    drawing the ellipse is to draw the major and
    minor axes perpendicular bisecting each
    other. The major and the minor axes are
    perpendicular bisectors of each other.
  1. 10.Loop of the thread method is the practical
    application of method.

a) Oblong method
b) Trammel method
c) Arcs of circles method
d) Concentric method
Answer: c
Explanation:
Loop of the thread method is
the practical application of the arcs of circles
method. The lengths of the ends of the minor
axis are half of the length of the major axis.
In this method, a pin is inserted at the foci
point and the thread is tied to a pencil which
is used to draw the curve.

CONSTRUCTION OF PARABOLA

  1. 1.Which of the following is incorrect about
    Parabola?

    a) Eccentricity is less than 1
    b) Mathematical equation is x2 = 4ay
    c) Length of latus rectum is 4a
    d) The distance from the focus to a vertex is
    equal to the perpendicular distance from a
    vertex to the directrix
    Answer: a
    Explanation:
    The eccentricity is equal to one.
    That is the ratio of a perpendicular distance
    from point on curve to directrix is equal to
    distance from point to focus. The eccentricity
    is less than 1 for an ellipse, greater than one
    for hyperbola, zero for a circle, one for a
    parabola.
  2. 2.Which of the following constructions use
    parabolic curves?

    a) Cooling towers
    b) Water channels
    c) Light reflectors
    d) Man-holes
    Answer: c
    Explanation:
    Arches, Bridges, sound
    reflectors, light reflectors etc use parabolic
    curves. Cooling towers, water channels use
    Hyperbolic curves as their design. Arches,
    bridges, dams, monuments, man-holes, glands
    and stuffing boxes etc use elliptical curves.
  3. 3.The length of the latus rectum of the
    parabola y2 =ax is

    a) 4a
    b) a
    c) a/4
    d) 2a
    Answer: b
    Explanation:
    Latus rectum is the line
    perpendicular to axis and passing through
    focus ends touching parabola. Length of latus
    rectum of y2 =4ax, x2 =4ay is 4a; y2 =2ax, x2=2ay is 2a; y2 =ax, x2 =ay is a.
  4. 4.Which of the following is not a parabola
    equation?

    a) x2 = 4ay
    b) y2– 8ax = 0
    c) x2 = by
    d) x2 = 4ay2
    Answer: d
    Explanation:
    The remaining represents
    different forms of parabola just by adjusting
    them we can get general notation of parabola
    but x2 = 4ay2 gives equation for hyperbola.
    And x2 + 4ay2 =1 gives equation for ellipse.
  5. 5.The parabola x2 = ay is symmetric about x-axis.
    a) True
    b) False
    Answer: b
    Explanation:
    From the given parabolic
    equation x2 = ay we can easily say if we give
    y values to that equation we get two values
    for x so the given parabola is symmetric
    about y-axis. If the equation is y2 = ax then it
    is symmetric about x-axis.
  1. 6.Steps are given to find the axis of a
    parabola. Arrange the steps.

i. Draw a perpendicular GH to EF which cuts
parabola.
ii. Draw AB and CD parallel chords to given
parabola at some distance apart from each
other.
iii. The perpendicular bisector of GH gives
axis of that parabola.
iv. Draw a line EF joining the midpoints lo
AB and CD.
a) i, ii, iii, iv
b) ii, iv, i, iii
c) iii, iv, i, ii
d) iv, i, ii, iii
Answer: b
Explanation:
First we drawn the parallel
chords and then line joining the midpoints of
the previous lines which is parallel to axis so
we drawn the perpendicular to this line and
then perpendicular bisector gives the axis of
parabola.

  1. 7.Steps are given to find focus for a parabola.
    Arrange the steps.

    i. Draw a perpendicular bisector EF to BP,
    Intersecting the axis at a point F.
    ii. Then F is the focus of parabola.
    iii. Mark any point P on the parabola and
    draw a perpendicular PA to the axis.
    iv. Mark a point B on the on the axis such that
    BV = VA (V is vertex of parabola). Join B
    and P.
    a) i, ii, iii, iv
    b) ii, iv, i, iii
    c) iii, iv, i, ii
    d) iv, i, ii, iii
    Answer: c
    Explanation:
    Initially we took a parabola
    with axis took any point on it drawn a
    perpendicular to axis. And from the point
    perpendicular meets the axis another point is
    taken such that the vertex is equidistant from
    before point and later point. Then from that
    one to point on parabola a line is drawn and
    perpendicular bisector for that line meets the
    axis at focus.
  2. 8.Which of the following is not belonged to
    ellipse?

    a) Latus rectum
    b) Directrix
    c) Major axis
    d) Axis
    Answer: c
    Explanation:
    Latus rectum is the line joining
    one of the foci and perpendicular to the major
    axis. Major axis and minor axis are in ellipse
    but in parabola, only one focus and one axis
    exist since eccentricity is equal to 1.

CONSTRUCTION OF HYPERBOLA

  1. 1.Which of the following is Hyperbola
    equation?

    a) y2 + x2/b2 = 1
    b) x2= 1ay
    c)x2 /a2– y2/b2 = 1
    d) x2 + y2 = 1
    Answer: c
    Explanation:
    The equation x2+ y2 = 1 gives
    a circle; if the x2 and y2 have same co-efficient then the equation gives circles. The
    equation x2= 1ay gives a parabola. The
    equation y2 + x2/b2 = 1 gives an ellipse.
  2. 2.Which of the following constructions use
    hyperbolic curves?

    a) Cooling towers
    b) Dams
    c) Bridges
    d) Man-holes
    Answer: a
    Explanation:
    Cooling towers, water channels
    use Hyperbolic curves as their design.
    Arches, Bridges, sound reflectors, light
    reflectors etc., use parabolic curves. Arches,
    bridges, dams, monuments, man-holes, glands
    and stuffing boxes etc., use elliptical curves.
  1. 3.The lines which touch the hyperbola at an
    infinite distance are

    a) Axes
    b) Tangents at vertex
    c) Latus rectum
    d) Asymptotes
    Answer: d
    Explanation:
    Axis is a line passing through
    the focuses of a hyperbola. The line which
    passes through the focus and perpendicular to
    the major axis is latus rectum. Tangent is the
    line which touches the curve at only one
    point.
  2. 4.Which of the following is the eccentricity
    for hyperbola?

    a) 1
    b) 3/2
    c) 2/3
    d) 1/2
    Answer: b
    Explanation:
    The eccentricity for an ellipse
    is always less than 1. The eccentricity is
    always 1 for any parabola. The eccentricity is
    always 0 for a circle. The eccentricity for a
    hyperbola is always greater than 1.
  3. 5.If the asymptotes are perpendicular to each
    other then the hyperbola is called rectangular
    hyperbola.

    a) True
    b) False
    Answer: a
    Explanation:
    In ellipse there exist two axes
    (major and minor) which are perpendicular to
    each other, whose extremes have tangents
    parallel them. There exist two conjugate axes
    for ellipse and 1 for parabola and hyperbola.
  4. 6.A straight line parallel to asymptote
    intersects the hyperbola at only one point.

    a) True
    b) False
    Answer: a
    Explanation:
    A straight line parallel to
    asymptote intersects the hyperbola at only
    one point. This says that the part of hyperbola
    will lay in between the parallel lines through
    outs its length after intersecting at one point.
  5. 7.Steps are given to locate the directrix of
    hyperbola when axis and foci are given.
    Arrange the steps.

    i. Draw a line joining A with the other Focus
    F.
    ii. Draw the bisector of angle FAF1, cutting
    the axis at a point B.
    iii. Perpendicular to axis at B gives directrix.
    iv. From the first focus F1 draw a
    perpendicular to touch hyperbola at A.
    a) i, ii, iii, iv
    b) ii, iv, i, iii
    c) iii, iv, i, ii
    d) iv, i, ii, iii
    Answer: d
    Explanation:
    The directrix cut the axis at the
    point of intersection of the angular bisector of
    lines passing through the foci and any point
    on a hyperbola. Just by knowing this we can
    find the directrix just by drawing
    perpendicular at that point to axis.
  6. 8.Steps are given to locate asymptotes of
    hyperbola if its axis and focus are given.
    Arrange the steps.

    i. Draw a perpendicular AB to axis at vertex.
    ii. OG and OE are required asymptotes.
    iii. With O midpoint of axis (centre) taking
    radius as OF (F is focus) draw arcs cutting
    AB at E, G.
    iv.Join O, G and O, E.
    a) i, iii, iv, ii
    b) ii, iv, i, iii
    c) iii, iv, i, ii
    d) iv, i, ii, iii
    Answer: b
    Explanation:
    Asymptotes pass through centre is the main point and then the asymptotes cut the directrix and perpendiculars at focus are known and simple. Next comes is where the asymptotes cuts the perpendiculars, it is at distance of centre to vertex and centre to focus respectively.


  1. 9.The asymptotes of any hyperbola intersects at
    a) On the directrix
    b) On the axis
    c) At focus
    d) Centre
    Answer: d
    Explanation:
    The asymptotes intersect at
    centre that is a midpoint of axis even for
    conjugate axis it is valid. Along with the
    hyperbola asymptotes are also symmetric
    about both axes so they should meet at centre
    only.

CONSTRUCTION OF CYCLOIDAL CURVES

  1. 1.Is a curve generated by a
    point fixed to a circle, within or outside its
    circumference, as the circle rolls along a
    straight line.

    a) Cycloid
    b) Epicycloid
    c) Epitrochoid
    d) Trochoid
    Answer: d
    Explanation:
    Cycloid form if generating
    point is on the circumference of generating a
    circle. Epicycloid represents generating circle
    rolls on the directing circle. Epitrochoid is
    that the generating point is within or outside
    the generating circle but generating circle
    rolls on directing circle.
  2. 2.Is a curve generated by a
    point on the circumference of a circle, which
    rolls without slipping along another circle
    outside it.

    a) Trochoid
    b) Epicycloid
    c) Hypotrochoid
    d) Involute
    Answer: b
    Explanation:
    Trochoid is curve generated by
    a point fixed to a circle, within or outside its
    circumference, as the circle rolls along a
    straight line. ‘Hypo’ represents the generating
    circle is inside the directing circle.
  3. 3.Is a curve generated by a point
    on the circumference of a circle which rolls
    without slipping on a straight line.

    a) Trochoid
    b) Epicycloid
    c) Cycloid
    d) Evolute
    Answer: c
    Explanation:
    Trochoid is curve generated by
    a point fixed to a circle, within or outside its
    circumference, as the circle rolls along a
    straight line. Cycloid is a curve generated by
    a point on the circumference of a circle which
    rolls along a straight line. ‘Epi’ represents the
    directing path is a circle.
  4. 4.When the circle rolls along another circle
    inside it, the curve is called a

    a) Epicycloid
    b) Cycloid
    c) Trochoid
    d) Hypocycloid
    Answer: d
    Explanation:
    Cycloid is a curve generated by
    a point on the circumference of a circle which
    rolls along a straight line. ‘Epi’ represents the
    directing path is a circle. Trochoid is a curve
    generated by a point fixed to a circle, within
    or outside its circumference, as the circle rolls
    along a straight line. ‘Hypo’ represents the
    generating circle is inside the directing circle.
  1. 6.Match the following
  2. Generating point is – i. Inferior within the circumference trochoid

Generating point is on the circumference of circle and – ii. Epicycloid generating circle rolls on
straight line.

The circumference of circle and generating circle rolls on straight line – iii. Cycloid

Generating point is on the circumference of circle

and generating circle rolls – iv. Superior
trochoid

a) 1, i; 2, iii; 3, iv; 4, ii
b) 1, ii; 2, iii; 3, i; 4, iv
c) 1, ii; 2, iv; 3, iii; 4, i
d) 1, iv; 2, iii; 3, ii; 4, i
Answer: a

Explanation: Trochoid is curve generated by
a point fixed to a circle, within or outside its
circumference, as the circle rolls along a
straight line. Inferior or superior depends on
whether the generating point in within or
outside the generating circle. If directing path
is straight line then the curve is cycloid.

  1. 7.Steps are given to find the normal and
    tangent for a cycloid. Arrange the steps if C is
    the centre for generating circle and PA is the
    directing line. N is the point on cycloid.

    i. Through M, draw a line MO perpendicular
    to the directing line PA and cutting at O.
    ii. With centre N and radius equal to radius of
    generating circle, draw an arc cutting locus of
    C at M.

iii. Draw a perpendicular to ON at N which is tangent.

iv. Draw a line joining O and N which is
normal.

a) iii, i, iv, ii
b) ii, i, iv, iii
c) iv, ii, i, iii
d) i, iv, iii, ii
Answer: b
Explanation:
The normal at any point on a cycloidal curve will pass through the corresponding point of contact between the generating circle and the directing line. So with help of locus of centre of generating circle we found the normal and the tangent.

  1. 8.Steps are given to find the normal and
    tangent to an epicycloid. Arrange the steps if
    C is the centre for generating circle and O is
    the centre of directing cycle. N is the point on
    epicycloid.

    i. Draw a line through O and D cutting
    directing circle at M.
    ii. Draw perpendicular to MN at N. We get
    tangent.
    iii. With centre N and radius equal to radius
    of generating circle, draw an arc cutting the
    locus of C at D.
    iv. Draw a line joining M and N which is
    normal.

    a) iii, i, iv, ii
    b) ii, i, iv, iii
    c) iv, ii, i, iii
    d) i, iv, iii, ii
    Answer: a
    Explanation:
    The normal at any point on an
    epicycloidal curve will pass through the
    corresponding point of contact between the
    generating circle and the directing circle. And
    also with help of locus of centre of generating
    circle we found the normal and the tangent.
  2. 9.The generating circle will be inside the
    directing circle for

    a) Cycloid
    b) Inferior trochoid
    c) Inferior epitrochoid
    d) Hypocycloid
    Answer: d
    Explanation:
    The generating circle will be
    inside the directing circle for hypocycloid or
    hypotrochoid. Trochoid is a curve generated
    by a point fixed to a circle, within or outside
    its circumference, as the circle rolls along a
    straight line or over circle if not represented
    with hypo as a prefix.
  3. 10.The generating point is outside the
    generating circle for

    a) Cycloid
    b) Superior Trochoid
    c) Inferior Trochoid
    d) Epicycloid
    Answer: b
    Explanation:
    If the generating point is on the
    circumference of generating circle then the
    curve formed may be cycloids or
    hypocycloids. Trochoid is a curve generated
    by a point fixed to a circle, within or outside
    its circumference, as the circle rolls along a
    straight line or a circle. But here given is
    outside so it is superior trochoid.

CONSTRUCTION OF INVOLUTE

  1. 1.Mathematical equation for Involute is
    a) x = a cos3θ
    b) x = r cosθ + r θ sinθ
    c) x = (a+b)cosθ – a cos(a+b⁄a θ)
    d) y = a(1-cosθ)
    Answer: b
    Explanation:
    x= a cos3 Ɵ is equation forhypocycloid, x= (a+ b) cosƟ – a cos ((a+b)/aƟ) is equation for epicycloid, y= a (1-cosƟ) is equation for cycloid and x = r cosƟ r Ɵ sinƟ is equation for Involute.
  1. 2.Steps are given to draw involute of given
    circle. Arrange the steps f C is the centre of
    circle and P be the end of the thread (starting
    point).

    i. Draw a line PQ, tangent to the circle and
    equal to the circumference of the circle.
    ii. Draw the involute through the points P1,
    P2, P3 ……….etc.
    iii. Divide PQ and the circle into 12 equal
    parts.
    iv. Draw tangents at points 1, 2, 3 etc. and
    mark on them points P1, P2, P3 etc. such that
    1P1 =P1l, 2P2 = P2l, 3P3= P3l etc.
    a) ii, i, iv, iii

b) iii, i , iv, ii
c) i, iii, iv, ii
d) iv, iii, i, ii
Answer: c
Explanation:
Involute is a curve which is
formed by the thread which is yet complete a
single wound around a circular object so thus
the thread having length equal to the
circumference of the circular object. And the
involute curve follows only the thread is kept
straight while wounding.

  1. 3.Steps are given to draw tangent and normal
    to the involute of a circle (center is C) at a
    point N on it. Arrange the steps.

    i. With CN as diameter describe a semi-circle
    cutting the circle at M.
    ii. Draw a line joining C and N.
    iii. Draw a line perpendicular to NM and
    passing through N which is tangent.
    iv. Draw a line through N and M. This line is
    normal.
    a) ii, i, iv, iii
    b) iii, i , iv, ii
    c) i, iii, iv, ii
    d) iv, iii, i, ii
    Answer: a
    Explanation:
    The normal to an involute of a
    circle is tangent to that circle. So simply by
    finding the appreciable tangent of circle
    passing through the point given on involute
    gives the normal and then by drawing
    perpendicular we can find the tangent to
    involute.
  2. 4.Steps given are to draw an involute of a
    given square ABCD. Arrange the steps.
    i. With B as centre and radius BP1 (BA+ AD)
    draw an arc to cut the line CB-produced at
    P2.

    ii. The curve thus obtained is the involute of
    the square.
    iii. With centre A and radius AD, draw an arc
    to cut the line BA-produced at a point P1.
    iv. Similarly, with centres C and D and radii
    CP2 and DP3 respectively, draw arcs to cut
    DC-produced at P3 and AD-produced at P4.
    a) ii, i, iv, iii
    b) iii, i , iv, ii
    c) i, iii, iv, ii
    d) iv, iii, i, ii
    Answer: b
    Explanation:
    It is easy to draw involutes to
    polygons. First, we have to point the initial
    point and then extending the sides. Then
    cutting the extended lines with cumulative
    radiuses of length of sides gives the points on
    involute and then joining them gives involute.
  3. 5.Steps given are to draw an involute of a
    given triangle ABC. Arrange the steps.

    i. With C as centre and radius C1 draw arc
    cutting AC-extended at 2.
    ii. With A as center and radius A2 draw an
    arc cutting BA- extended at 3 completing
    involute.
    iii. B as centre with radius AB draw an arc
    cutting the BC- extended at 1.
    iv. Draw the given triangle with corners A, B, C.
    a) ii, i, iv, iii
    b) iii, i , iv, ii
    c) i, iii, iv, ii
    d) iv, iii, i, ii
    Answer: d
    Explanation:
    It will take few simple steps to
    draw involute for a triangle since it has only 3
    sides. First, we have to point the initial point
    and then extending the sides. Then cutting the
    extended lines with cumulative radiuses of
    length of sides gives the points on involute
    and then joining them gives involute.
  4. 6.Steps given are to draw an involute of a
    given pentagon ABCDE. Arrange the steps.
    i. B as centre and radius AB, draw an arc
    cutting BC –extended at 1.

    ii. The curve thus obtained is the involute of
    the pentagon.
    iii. C as centre and radius C1, draw an arc
    cutting CD extended at 2.
    iv. Similarly, D, E, A as centres and radiusD2, E3, A4, draw arcs cutting DE, EA, AB at 3, 4, 5 respectively.


a) ii, i, iv, iii
b) iii, i , iv, ii
c) i, iii, iv, ii
d) iv, iii, i, ii
Answer: c
Explanation:
It is easy to draw involutes to
polygons. First, we have to point the initial
point and then extending the sides. Then
cutting the extended lines with cumulative
radiuses of length of sides gives the points on
involute and then joining them gives involute.

  1. 7.For inferior trochoid or inferior epitrochoid
    the curve touches the directing line or
    directing circle.

    a) True
    b) False
    Answer: b
    Explanation:
    Since in the inferior trochoids
    the generating point is inside the generating
    circle the path will be at a distance from
    directing line or circle even if the generating
    circle is inside or outside the directing circle.
  2. 8.‘Hypo’ as prefix to cycloids give that the
    generating circle is inside the directing circle.

    a) True
    b) False
    Answer: a
    Explanation:
    ‘Hypo’ represents the
    generating circle is inside the directing circle.
    ‘Epi’ represents the directing path is a circle.
    Trochoid represents the generating point is
    not on the circumference of generating a
    circle.

CONSTRUCTION OF SPIRAL

  1. 1.Which of the following represents an
    Archemedian spiral?

    a) Tornado
    b) Cyclone
    c) Mosquito coil
    d) Fibonacci series
    Answer: c
    Explanation:
    Archemedian spiral is a curve
    traced out by a point moving in such a way
    that its movement towards or away from the
    pole is uniform with the increase of the
    vectorial angle from the starting line. It is
    generally used for teeth profiles of helical
    gears etc.
  2. 2.Steps are given to draw normal and tangent
    to an archemedian curve. Arrange the steps, if
    O is the center of curve and N is point on it.

    i. Through N, draw a line ST perpendicular to
    NM. ST is the tangent to the spiral.
    ii. Draw a line OM equal in length to the
    constant of the curve and perpendicular to
    NO.
    iii. Draw the line NM which is normal to the
    spiral.
    iv. Draw a line passing through the N and O
    which is radius vector.
    a) ii, iv, i, iii
    b) i, iv, iii, ii
    c) iv, ii, iii, i
    d) iii, i, iv, ii
    Answer: c
    Explanation:
    The normal to an archemedian
    spiral at any point is the hypotenuse of the
    right angled triangle having the other two
    sides equal in length to the radius vector at
    that point and the constant of the curve
    respectively.
  3. 3.Which of the following does not represents
    an Archemedian spiral?

    a) Coils in heater
    b) Tendrils
    c) Spring
    d) Cyclone
    Answer: d
    Explanation:
    Tendrils are a slender thread-like structures of a climbing plant, often growing in a spiral form. For cyclones the moving point won’t have constant velocity. The archemedian spirals have a constant increase in the length of a moving point. Spring is a helix.
  4. 4.Logarithmic spiral is also called
    Equiangular spiral.

    a) True
    b) False
    Answer: a
    Explanation:
    The logarithmic spiral is also
    known as equiangular spiral because of its
    property that the angle which the tangent at
    any point on the curve makes with the radius
    vector at that point is constant. The values of
    vectorial angles are in arithmetical
    progression.
  5. In logarithmic Spiral, the radius vectors are
    in arithmetical progression.

    a) True
    b) False
    Answer: b
    Explanation:
    In the logarithmic Spiral, the
    values of vectorial angles are in arithmetical
    progression and radius vectors are in the
    geometrical progression that is the lengths of
    consecutive radius vectors enclosing equal
    angles are always constant.
  6. The mosquito coil we generally see in
    house hold purposes and heating coils in
    electrical heater etc are generally which
    spiral.

    a) Logarithmic spiral
    b) Equiangular spiral
    c) Fibonacci spiral
    d) Archemedian spiral
    Answer: d
    Explanation:
    Archemedian spiral is a curve
    traced out by a point moving in such a way
    that its movement towards or away from the
    pole is uniform with the increase of the
    vectorial angle from the starting line. The use
    of this curve is made in teeth profiles of
    helical gears, profiles of cam etc.

BASICS OF CONIC SECTIONS – 1

  1. 1.The sections cut by a plane on a right
    circular cone are called as

    a) Parabolic sections
    b) Conic sections
    c) Elliptical sections
    d) Hyperbolic sections
    Answer: b
    Explanation:
    The sections cut by a plane on a
    right circular cone are called as conic sections
    or conics. The plane cuts the cone on
    different angles with respect to the axis of the
    cone to produce different conic sections.
  2. 2.Which of the following is a conic section?
    a) Circle
    b) Rectangle
    c) Triangle
    d) Square
    Answer: a
    Explanation:
    Circle is a conic section. When
    the plane cuts the right circular cone at right
    angles with the axis of the cone, the shape
    obtained is called as a circle. If the angle is
    oblique we get the other parts of the conic
    sections.
  3. 3.In conics, the is revolving to form
    two anti-parallel cones joined at the apex.

    a) Ellipse
    b) Circle
    c) Generator
    d) Parabola
    Answer: c
    Explanation:
    In conics, the generator is
    revolving to form two anti-parallel cones
    joined at the apex. The plane is then made to
    cut these cones and we get different conic
    sections. If we cut at right angles with respect
    to the axis of the cone we get a circle.
  4. 4.While cutting, if the plane is at an angle
    and it cuts all the generators, then the conic
    formed is called as

    a) Circle
    b) Ellipse
    c) Parabola
    d) Hyperbola
    Answer: b
    Explanation:
    If the plane cuts all the
    generators and is at an angle to the axis of the
    cone, then the resulting conic section is called
    as an ellipse. If the cutting angle was right
    angle and the plane cuts all the generators
    then the conic formed would be circle.
  1. 5.If the plane cuts at an angle to the axis but
    does not cut all the generators then what is
    the name of the conics formed?

a) Ellipse
b) Hyperbola
c) Circle
d) Parabola
Answer: d
Explanation:
If the plane cuts at an angle
with respect to the axis and does not cut all
the generators then the conics formed is a
parabola. If the plane cuts all the generators
then the conic section formed is called as
ellipse.

  1. 6.When the plane cuts the cone at angle
    parallel to the axis of the cone, then is
    formed.

    a) Hyperbola
    b) Parabola
    c) Circle
    d) Ellipse
    Answer: a
    Explanation:
    When the plane cuts the cone at
    an angle parallel to the axis of the cone, then
    the resulting conic section is called as a
    hyperbola. If the plane cuts the cone at an
    angle with respect to the axis of the cone then
    the resulting conic sections are called as
    ellipse and parabola.
  2. 7.Which of the following is not a conic
    section?

    a) Apex
    b) Hyperbola
    c) Ellipse
    d) Parabola
    Answer: a
    Explanation:
    Conic sections are formed
    when a plane cuts through the cone at an
    angle with respect to the axis of the cone. If
    the angle is right angle then the conics is a
    circle, if the angle is oblique then the
    resulting conics are parabola and ellipse.
  3. 8.The locus of point moving in a plane such
    that the distance between a fixed point and a
    fixed straight line is constant is called as

    a) Conic
    b) Rectangle
    c) Square
    d) Polygon
    Answer: a
    Explanation:
    The locus of a point moving in
    a plane such that the distance between a fixed
    point and a fixed straight line is always
    constant. The fixed straight line is called as
    directrix and the fixed point is called as the
    focus.
  4. 9.The ratio of the distance from the focus to
    the distance from the directrix is called as
    eccentricity.

    a) True
    b) False
    Answer: a
    Explanation:
    The ratio of the distance from
    the focus to the distance from the directrix is
    called eccentricity. It is denoted as e. The
    value of eccentricity can give information
    regarding which type of conics it is.
  5. 10.Which of the following conics has an
    eccentricity of unity?

    a) Circle
    b) Parabola
    c) Hyperbola
    d) Ellipse
    Answer: b
    Explanation:
    Eccentricity is defined as the
    ratio of the distance from the focus to the
    distance from the directrix. It is denoted as e.
    The value of eccentricity can give
    information regarding which type of conics it
    is. The eccentricity of a parabola is the unity
    that is 1.
  6. Which of the following has an
    eccentricity less than one?

    a) Circle
    b) Parabola

c) Hyperbola
d) Ellipse
Answer: d
Explanation:
Eccentricity is defined as the
ratio of the distance from the focus to the
distance from the directrix. It is denoted as e.
The value of eccentricity can give
information regarding which type of conics it
is. The eccentricity of an ellipse is less than
one.

  1. 12.If the distance from the focus is 10 units
    and the distance from the directrix is 30 units,
    then what is the eccentricity?

    a) 0.3333
    b) 0.8333
    c) 1.6667
    d) 0.0333
    Answer: a
    Explanation:
    Eccentricity is defined as the
    ratio of the distance from the focus to the
    distance from the directrix. Hence from the
    formula of eccentricity, e = 10 ÷ 30 = 0.3333.
    Since the value of eccentricity is less than one
    the conic is an ellipse.
  2. 13.If the value of eccentricity is 12, then
    what is the name of the conic?

    a) Ellipse
    b) Hyperbola
    c) Parabola
    d) Circle
    Answer: b
    Explanation:
    Eccentricity is defined as the
    ration of the distance from the focus to the
    distance from the directrix. It is denoted as e.
    If the value of eccentricity is greater than
    unity then the conic section is called as a
    hyperbola.
  3. 14.If the distance from the focus is 3 units
    and the distance from the directrix is 3 units,
    then how much is the eccentricity?

    a) Infinity
    b) Zero
    c) Unity
    d) Less than one
    Answer: c
    Explanation:
    Eccentricity is defined as the
    ration of the distance from the focus to the
    distance from the directrix and it is denoted as
    e. Hence from the definition, e = 3 ÷ 3 = 1.
    Hence the value of eccentricity is equal to
    unity.
  4. 15.If the distance from the focus is 2 mm and
    the distance from the directrix is 0.5 mm then
    what is the name of the conic section?

    a) Circle
    b) Ellipse
    c) Parabola
    d) Hyperbola
    Answer: d
    Explanation:
    The eccentricity is defined as
    the ratio of the distance from the focus to the
    distance from the directrix. It is denoted as e.
    If the value of the eccentricity is greater than
    unity then the conic section is called as a
    hyperbola.

BASICS OF CONIC SECTIONS – 2

  1. 1.Which of the following is a conic section?
    a) Apex
    b) Circle
    c) Rectangle
    d) Square
    Answer: b
    Explanation:
    Conic sections are formed
    when a plane cuts through the cone at an
    angle with respect to the axis of the cone. If
    the angle is right angle then the conics is a
    circle, if the angle is oblique then the
    resulting conics are parabola and ellipse.
  2. 2.Which of the following has an eccentricity
    more than unity?

    a) Parabola

b) Circle
c) Hyperbola
d) Ellipse
Answer: c
Explanation:
Eccentricity is defined as the
ratio of the distance from the focus to the
distance from the directrix. It is denoted as e.
The value of eccentricity can give
information regarding which type of conics it
is. The eccentricity of a hyperbola is more
than one.

  1. 3.If the distance from the focus is 10 units
    and the distance from the directrix is 30 units,
    then what is the name of the conic?

    a) Circle
    b) Parabola
    c) Hyperbola
    d) Ellipse
    Answer: d
    Explanation:
    Eccentricity is defined as the
    ratio of the distance from the focus to the
    distance from the directrix. Hence from the
    formula of eccentricity, e = 10 ÷ 30 = 0.3333.
    Since the value of eccentricity is less than one
    the conic is an ellipse.
  2. 4.If the distance from the focus is 2 mm and
    the distance from the directrix is 0.5 mm then
    what is the value of eccentricity?

    a) 0.4
    b) 4
    c) 0.04
    d) 40
    Answer: b
    Explanation:
    Eccentricity is defined as the
    ratio of the distance from the focus to the
    distance from the directrix and it is denoted
    by e. Therefore, by definition, e = 2 ÷ 0.5 = 4.
    Hence the conic section is called as
    hyperbola.
  3. 5.If the distance from the focus is 3 units and
    the distance from the directrix is 3 units, then
    what is the name of the conic section?

    a) Ellipse
    b) Hyperbola
    c) Circle
    d) Parabola
    Answer: d
    Explanation:
    Eccentricity is defined as the
    ratio of the distance from the focus to the
    distance from the directrix and it is denoted
    by e. Therefore, by definition, e = 3 ÷ 3 = 1.
    Hence the conic section is called as a
    parabola.
  4. 6.If the distance from the directrix is 5 units
    and the distance from the focus is 3 units then
    what is the name of the conic section?

    a) Ellipse
    b) Parabola
    c) Hyperbola
    d) Circle
    Answer: a
    Explanation:
    Eccentricity is defined as the
    ratio of the distance from the focus to the
    distance from the directrix and it is denoted
    by e. Hence, by definition, e = 3 ÷ 5 = 0.6.
    Hence the conic section is called an ellipse.
  5. 7.If the distance from a fixed point is greater
    than the distance from a fixed straight line
    then what is the name of the conic section?

    a) Parabola
    b) Circle
    c) Hyperbola
    d) Ellipse
    Answer: c
    Explanation:
    The fixed point is called as
    focus and the fixed straight line is called as
    directrix. Eccentricity is defined as the ratio
    of the distance from the focus to the distance
    from the directrix and it is denoted by e. If e
    is greater than one then the conic section is
    called as a hyperbola.
  6. 8.If the distance from a fixed straight line is
    equal to the distance from a fixed point then
    what is the name of the conic section?

b) Parabola
c) Hyperbola
d) Circle
Answer: b
Explanation:
The fixed straight line is called
as directrix and the fixed point is called as a
focus. Eccentricity is defined as the ratio of
the distance from the focus to the distance
from the directrix and it is denoted by e.
Eccentricity of a parabola is unity.

  1. 9.If the distance from the directrix is greater
    than the distance from the focus then what is
    the value of eccentricity?

    a) Unity
    b) Less than one
    c) Greater than one
    d) Zero
    Answer: b
    Explanation:
    Eccentricity is defined as the
    ratio of the distance from the focus to the
    distance from the directrix and it is denoted
    by e. Therefore, by definition the value of
    eccentricity is less than one hence the conic
    section is an ellipse.
  2. 10.If the distance from the directrix is 5 units
    and the distance from the focus is 3 units then
    what is the value of eccentricity?

    a) 1.667
    b) 0.833
    c) 0.60
    d) 0.667
    Answer: c
    Explanation:
    Eccentricity is defined as the
    ratio of the distance from the focus to the
    distance from the directrix and it is denoted
    by e. Therefore, by definition, e = 3 ÷ 5 = 0.6.
    Hence the conic section is called an ellipse.
  3. 11.If the distance from a fixed straight line is
    5mm and the distance from a fixed point is
    14mm then what is the name of the conic
    section?

    a) Hyperbola
    b) Parabola
    c) Ellipse
    d) Circle
    Answer: a
    Explanation:
    The fixed straight line is called
    directrix and the fixed point is called as a
    focus. Eccentricity is defined as the ratio of
    the distance from the focus to the distance
    from the directrix and it is denoted by e.
    Hence from definition e = 14 ÷ 5 = 2.8. The
    eccentricity of a hyperbola is greater than
    one.
  4. 12.If the distance from the directrix is greater
    than the distance from the focus then what is
    the name of the conic section?

    a) Hyperbola
    b) Parabola
    c) Ellipse
    d) Circle
    Answer: c
    Explanation:
    Eccentricity is defined as the
    ratio of the distance from the focus to the
    distance from the directrix and it is denoted
    by e. Therefore, by definition the value of
    eccentricity is less than one hence the conic
    section is an ellipse.
  5. 13.If the distance from a fixed straight line is
    equal to the distance from a fixed point then
    what is the value of eccentricity?

    a) Unity
    b) Greater than one
    c) Infinity
    d) Zero
    Answer: a
    Explanation:
    The fixed straight line is called
    as directrix and the fixed point is called as a
    focus. Eccentricity is defined as the ratio of
    the distance from the focus to the distance
    from the directrix and it is denoted by e.
    Hence from definition e = x ÷ x = 1.
  1. 14.If the distance from a fixed point is
    greater than the distance from a fixed straight
    line then what is the value of eccentricity?

    a) Unity
    b) Infinity
    c) Zero
    d) Greater than one
    Answer: d
    Explanation:
    The fixed point is called as
    focus and the fixed straight line is called as
    directrix. Eccentricity is defined as the ratio
    of the distance from the focus to the distance
    from the directrix and it is denoted by e.
    Hence from the definition, the value of
    eccentricity is greater than one.
  2. 15.If the distance from a fixed straight line is
    5mm and the distance from a fixed point is
    14mm then what is the value of eccentricity?

    a) 0.357
    b) 3.57
    c) 2.8
    d) 0.28
    Answer: c
    Explanation:
    The fixed straight line is called
    as directrix and the fixed point is called as a
    focus. Eccentricity is defined as the ratio of
    the distance from the focus to the distance
    from the directrix and it is denoted by e.
    Hence from definition e = 14 ÷ 5 = 2.8.

BASICS OF CONIC SECTIONS – 3

1.Choose the correct option.

(a)Eccentricity= Distance of the point from the focus/ Distance of the point from the vertex

(b)Eccentricity= Distance of the point from the focus/ Distance of the point from the directrix

(c)Eccentricity= Distance of the point from the directrix / Distance of the point from the focus

(d)Eccentricity= Distance of the point from the latus rectum/ Distance of the point from the focus

Answer: b
Explanation:
The point where the extension
of major axis meets the curve is called vertex.
The conic is defined as the locus of a point in
such a way that the ratio of its distance from a
fixed point and a fixed straight line is always
constant. The ratio gives the eccentricity. The
fixed point is called the focus and the fixed
line is called directrix.

2. 2.Match the following.

A. E < 1 i. Rectangular hyperbola

B. E = 1 ii. Hyperbola

C. E > 1 iii. Ellipse

D. E > 1 iv. Parabola

a) A, i; B, ii; C, iii; D, iv

b) A, ii; B, iii; C, iv; D, i

c) A, iii; B, iv; C, ii; D, i

d) A, iv; B, iii; C, ii; D, i

Answer: c

Explanation: The conic is defined as the locus of a point in such a way that the ratio of its distance from a fixed point and a fixed straight line is always constant. The fixed point is called the focus and the fixed line is called directrix. The change in ratio as given above results in different curves.

  1. 3.A plane is parallel to a base of regular cone
    and cuts at the middle. The cross-section is

a) Circle
b) Parabola
c) Hyperbola
d) Ellipse
Answer: a
Explanation:
A cone is formed by reducing
the cross-section of a circle the point. So
there exist circles along the cone parallel to
the base. Since the given plane is parallel to
the base of the regular cone. The crosssection will be circle

  1. 4.The cross-section is a when a
    plane is inclined to the axis and cuts all the
    generators of a regular cone.

    a) Rectangular Hyperbola
    b) Hyperbola
    c) Circle
    d) Ellipse
    Answer: d
    Explanation:
    A cone is a solid or hollow
    object which tapers from a circular base to a
    point. Here given an inclined plane which
    cuts all the generators of a regular cone. So
    the cross-section will definitely ellipse.
  2. 5.The curve formed when eccentricity is
    equal to one is

    a) Parabola
    b) Circle
    c) Semi-circle
    d) Hyperbola
    Answer: a
    Explanation:
    The answer is parabola. Circle
    has an eccentricity of zero and semi circle is
    part of circle and hyper eccentricity is greater
    than one.
  1. 6.The cross-section gives a when the cutting plane is parallel to axis of cone.
    a) Parabola
    b) Hyperbola
    c) Circle
    d) Ellipse
    Answer: b
    Explanation:
    If the cutting plane makes angle
    less than exterior angle of the cone the crosssection gives a ellipse. If the cutting plane
    makes angle greater than the exterior angle of
    the cone the cross- section may be parabola or
    hyperbola.
  1. 7.A plane cuts the cylinder the plane is not
    parallel to the base and cuts all the generators.
    The Cross-section is

    a) Circle
    b) Ellipse
    c) Parabola
    d) Hyperbola
    Answer: b
    Explanation:
    Given is a plane which is
    inclined but cutting all the generators so it
    will be an ellipse. Cutting of all generators
    gives us information that the cross-section
    will be closed curve and not parabola or
    hyperbola. Circle will form only if plane is
    parallel to the base.
  2. 8.A plane cuts the cylinder and the plane is
    parallel to the base and cuts all the generators.
    The Cross-section is

    a) Circle
    b) Ellipse
    c) Parabola
    d) Rectangular hyperbola
    Answer: a
    Explanation:
    The plane which is parallel to base will definitely cut the cone at all generators. Here additional information also given that the plane is parallel to base so the cross-section will be circle.
  1. 9.The curve which has eccentricity zero is
    a) Parabola
    b) Ellipse
    c) Hyperbola
    d) Circle
    Answer: d
    Explanation:
    The eccentricity is the ratio of a
    distance from a point on the curve to focus
    and to distance from the point to directrix.
    For parabola it is 1 and for ellipse it is less
    than 1 and for hyperbola it is greater than 1.
    And for circle it is zero.
  2. 10.Rectangular hyperbola is one of the
    hyperbola but the asymptotes are
    perpendicular in case of rectangular
    hyperbola.

    a) True
    b) False
    Answer: a
    Explanation:
    Asymptotes are the tangents
    which meet the curve hyperbola at infinite
    distance. If the asymptotes are perpendicular
    to each other then hyperbola takes the name
    of a rectangular hyperbola.

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